View source for Right group

A semi-group which is right simple (cf. [[Simple semi-group|Simple semi-group]]) and satisfies the left cancellation law. Every right group is a [[Completely-simple semi-group|completely-simple semi-group]]. The property that a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082360/r0823601.png" /> is a right group is equivalent to any of the following conditions: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082360/r0823602.png" /> is right simple and contains an idempotent element; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082360/r0823603.png" /> is regular (cf. [[Regular semi-group|Regular semi-group]]) and satisfies the left cancellation law; c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082360/r0823604.png" /> can be partitioned into left ideals which are (necessarily isomorphic) groups; and d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082360/r0823605.png" /> is the direct product of a group and a right zero semi-group (cf. [[Idempotents, semi-group of|Idempotents, semi-group of]]). The notion of a left group is similar to that of a right group. Only groups are simultaneously right groups and left groups. Every completely-simple semi-group can be partitioned into right (left) ideals which are (necessarily isomorphic) right (left) groups.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,   G.B. Preston,   "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR></table>